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Nates on Groups and Geometry, 19 78" 1986 
by Steven H. Cullinane 

Typewritten notes coHected in a 40-page PDF document. 

CONTENTS 

01 1978-??-?? "An inrariance of symmetry*' Research announcement- 
OS 1978-10-?? "Symmetry invariance in a diamond ring," AXIS abstract 
received Oct.. 31, 1978. 

03 1978-12-?? "Orthogonality of Latin squares viewed as as skewness of lines" 

04 1981-11-05 "Patterns invariant, modulo rigid motions, under groups of 

discontinuous transformations- two examples" 

05 1981-12-24 "Solid symmetry"-- "Motions of each cell induce 

motions of the entire pattern." 

06 1982-05-12 "Map systems" Definition. 

07 1982-06-12 "I nscapes" Definition. 

08 1982-09-12 "A symplectic array" 

09 1982-09-22 "Inscapes II" Generalized definition. 

10 1982-12-27 "Group scores" Definition. 

11 1983-05-31 "Decompositions of group enveloping algebras* 1 

12 1983-06-21 "An invariance of symmetry* 1 Group actions on a 4x4x4 cube. 

13 1983-08-04 "Group identity algebras" Definition. 

14 1982-08-26 "Transformations over a bridge" Definition and problem. 

15 1983-10-01 "Portrait of O" Action of the octahedral group. 

16 1983-10-16 "Study of O" Group actions on two-color cubes. 

17 1983-11-08 " Compound groups* B efinition and problem. 

18 1983-11-10 "Group compounds" Definition and problem. 

19 1983-11-27 "Table groups" Definition and problem. 

20 1984-01-05 "Line ar op er at or s in ge ometric function spaces" 

Definition, theorem, problem. 

21 1984-09-15 "Diamonds and whirls" Blocks illustrate group actions. 

22 1984-09-25 "Affine groups on small binary spaces" Theorem. 

23 1985-03-26 "Visualizing GL(2,pV' Example on a 3x3 array. 

24 1985-04-05 "GL(2,3) actions on a cube" 

25 1985-04-05 "Group actions on partitions" Definition and problem. 

26 1985-04-28 "Generating the octad generator" 

27 1985-08-22 "Symmetry invariance under M12-" Theorem. 

28 1985-11-17 "A group bridge" Definition and problem. 

29 1985-12-11 "Dynamic and algebraic compatibility of groups" Problems. 

30 1986-01-11 " Ge ometry of p artitions II " A foray into analysis. 

31 1986-02-04 "Inscapes III: PG(2,4) from PG(.3,2)" 

32 1986-02-20 "The relativity problem in finite geometry" 

33 1986-03-31 "Group topologies" Definition and problems. 

34 1986-04-26 "Picturing the smallest projective 3-space" 

35 1986-05-08 "A linear complex related to M2 4" 

36 1986-05-26 "The 2-subsets of a 6-set are the points of a PG(3,2)" 

37 1986-06-26 "21 projective partitions" A 6-set model of PG(2,4). 

38 1 9 8 6 - 06- 1 1 "An out er automorp hism of S 6 relate d to M24" 

39 1986-07-03 " Picturing outer automorp hisms of S 6 " 

40 1986-07-11 "Inscapes IV- Inner and outer group actions" 

Web page URL,- http7;finitegeomet:iyorg/sc/gen/typednotes.html 



AH 1HVAHIANCE 07 SYMMETRY 
m 3TEVEJI E. OUIilHANE 



(Written sometime in 1978.) 



We present a simple, surprising, and beautiful combinatorial 
Invariants or geometric symmetry, *0 an algebraic setting* 

DEFINITION, A delta trgn_afoim of a square array over a U.-set is 
any pattern obtained from the array by a 1-to-l substitution or the 
four diagonally-divided t*C-color unit equates for tha if-sot elemonts. 



EXAMPLES. 



ffl-fflHSSSH .... 



PHEOSSIL. Every delta transform of the Klein group table has 
ordinary or col or- interchange symmetry, and remains symmetric under 
the group Qt Of y22. % $hfo transformations generated by combining 
permutation a of rows and columns with permutations of -quadrants* 



EXAMPLE. 




■fl 



^ 



$ 
o 



^ 



M 



PROOF (Sketch). The Klein group ia the additive group of CF(^); 
tbls suggests t*e regard the group's table T as a matrix over that 
field. So regarded, T is a linear combination of three (0^1 ^matrices 
that indicate the locations, lr. T, of the £-subsets of field elements. 
The structural symmetry of these matrices accounts for the ayrnnatpy 
of the delta transforms of T^ and is Invariant ■under C 

All delta transforms of the ty? matrices in the algebra generated by 
the images of T under fi are symmetric; there are many such algebras. 

THEOREM. If l<m±n z + 2 t there is an algebra of Ij.™ 
2n 7. 2ti matrices over GF(lj) with all delta transforms syrueetrie* 

An Induction proof constructs sets of basis matrices that yield 
the desired sysjuetry and ensure closure under multipll eats ion. 

REFERENCE 

S. H. Cullinane, Diamond theory (preprint)* 



Notices of the American Mathematical Society, February 1979! 
Issue 192, Volume 26, Number 2., pages A-193-A-194: 



Tyr-JOT 



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Etfarj -C-tHui e[ P bat ho* ordinal? o* color- Lrn arching* aymeerr. 



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**lnf >ni I nt mh pi *■() : f a tlnj nC * , Dl* rfaarlTli pattern!, Th*Tl ll H Inflrill* ruUr of HMK 

"Jlunhd" rtnma. Hflajorphle to rldii u( ■■trlraa ov» T »f*l- (Jtacalnd October 31, 1T7B. ! 



STEVElf H. CTLLBKH5 . 

Orthogonality Of Latin squares flawed fl.a a]cow»CBE of l ingj^. 

Shown below is a way to urcibcd tho six ordcr-U Latin squares that iiavp 
orthogonal Latin mates in a oat of 35 arrays so that orthogonality in the 
sot Of arrays corregpondfl to skeTmeaa in the act of 35 Una a of ?G{3»2}, 

Each array yioldB a. 3-set of diagrams that Show the Urea separating 
conrplamantary E-aiiDaeta of {0,,5»2,3} * each diagranq is the symmetric 
difference of the other two. The 3-aata of dlagraias correspond to tha 
lines dT FG(3,2}. Two arrays ara orthogonal Iff ■ their 3-aots or diagrams 
are disjoint., i.e. iff the corresponding lines of FGO r 2) eto aitcm. 

Thla is a new way of -slewing orth.CE on allty ° r Latin aqiiaraa, quite 
different frora their relationship to projectivo plane a. 

PHDBLEW: To what extent can this result be general laud 7 {Doe. 19?S ) 



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Steven 11. Cull, inane 

Patterns .i nvar iian; i, modulo rigid motions, uncer c?roat>s of aiEcontinuog g 

transfoiTiittons: lyo examples , Rtseatch i-i&fc<s. November 5, 1981.. 

In fig- 1, rigid motions of each cell in a pattern induce rigitf motions 
of tne entire pattern. In fig. 3, permutations of tells proc^ce various 
sectAOival vtevs of the sarna (modulo field motions) infinite plane pattern. 
These permutations are derived, as in fig. 2, item motions of a cub*. 




Fifi. 1 



ft* 
DC 
£ F 

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H o 




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FIS- 3 



en S, Cull inane 

jty^inBtrv . Expository note. December 24, L9fll. 



In pattern A, motions of each cell induce motions of 
the entire pattern; li.fcew3.se in B. 





B 



Steven H- Cullinane 

Mao systems . Query, May 12, 1982. 



Definition: Suppose every map d) into a given ring « can be 
vritten aa <f K ^ £;(*,.' 4*) . inhere N is a fixed positive 
integer, the e, are fixed elements of M, and the 0^ are fixed 
functions from M to P, b proper subset of M. 

Let A » {« *it > , = (* 1 . ... , IgJ-. 

The quintuple (M, A, P r 0, B) is a map system . 

Example: Using hexadecimal labels for the elements of GFU6] , 
l«t {H. A, I?, 0, H) - (GF(U}, A, ^O^B,^, f6, B, P) , 3). 
where the functions in A are specified by giving inverse images: 



10,1,3,21— iOl l0,4,C,Bl~i0l {0,5,*,A\-(Gt \ 

(C»D,F,E)~[B) (3.7,F,lM~»m U , 6.C9 J~l 9 } 
ifl^,B,A)-{9>, {2.6,E,A)*tEl, [2.7.D,8W7} / 



(*„«», *3 > 



{Multiplication in gf(1$J U here defined via the irreducible 

polynomial m 4 + X + l.J 

Remarks: Harmonic analysis allows a complicated map to be 
broken down into, or built Up from, simpler maps. 

Map systems are a different means to the same *nd. 

Query: What is known about such systems? 



Steven H. (Jul limine 

Inac&pfls . Query. June 12, l^ftS. 

Definition: Let R be an n-aiy symmetric relation on a set of 

t subsets of a t-eet, iftiere n< t -ut, for positive integers n,t,u,v. 

Represent eaoh Of the t subsets toy the l's In a nit array u^ 

over GF(£), There li l£ t. An In scape of R la a ujct array A of tho 

a^ flttah. that R is true for n of the a^_ {that is, for the subsets 

rapres anted by these a^) If and only if the arrangement of the a^ 

within A Is the same aa the arrangement of the l's In some a^ <£ . 

Examples: (light and dark represent O's and l r fl.) 



nun deeh 

HSEBBH HCBO 

-COCO ESDHn 

Remark a : Inscapes are useful for visualising relations in certain 

finite geometries . The above examples, for instance, illustrate 
relations among the 15 hyperplanea of ?G(3*2) and among the l£ lines 
fliad undar a particular symplsitlc polarity of F0(3i-2). 

Quepy: What is knoim about coaibinatorlal systems of trie sort? 

Note: For some other properties of the aj In the second example, 
sea B.F. Assaua, Jr. and J.E, Not5.Ho Sard.1, "Oari^ralizad 3fceiner 
ey stems of type 3-{*o", \ I).j6j jl] 1l » Finite Qeometrlea and Designs , 
London Hath. 3oe. Lecture Note Series h$ { Cambridge tinXv. Frese, 
Cambridge 198l}, pp. lb-Zl. 



Steven H* Cull in one 

A Bjmploctic array . Research not a. September 12, 19fl2. 

Tho 11x11 array below Is farmed by adding (light = G, dark = 1, 
1+1- 0) the 10 nonempty squares In the first column to the 10 
non&iupty squares in tho flrnt row. These squares represent tho 10 
pairs af lines interchanged mid or A particular oyuplectlc polarity 
Of PG{}*£}» The array Id of interest for several reasons ; 

1) It Sorven to Illustrate an eleavoatary, but useful, way of 
construct Lng a complicated combinatorial object from simpler 
objects: make an addition table. [ Closure Is not asaentlal.) 

2) Properties of array o thun for mod as addition tables may be of 
some Use in tho study Of 10x10 Latin squaree. 

3 J Site* each of the 121 Jp4* squares below ro pre a on t s a set of 
points In a finite projective space* the array may aorva to 
illustrate or to suggest properties of such 3pacoa. 

ncBBEannaHH 

□GHKHQCaSHaiH 

HBHtzsaaaaHE] 






8 



St o Ten H* Culiinnna 

In scapes II . Query, September £2 $ 1962. 

Given a set X of points, certain families- of subnets of X 
nnay havB, as families , Home property s» (Example: the families 
of spheres that are concentric, ) It may be that i?o- can associate 
to each point of X & subnet of X, via an Injection f:X-^2 , 
in ouch a way that the f-itaage, in turn, of this subset of X 
(i.e., the family of f- Imago a of it a points) is in fact one of 
the families of subsets of X that have property a. 
If the map f gives rise in this way to the sat & of a 11^ such 
s-fainill&B, we can write. In a cryptic but concise way, 
S=f£f(X)), and say that f la an In a cape of Si 

Query: "What knonn reaulta can he stated, after the appropriate 
definition of 3, in the form "There exists an inacape of S fl F 

Addendum of Oct. 10, 1982. 4 more precise definition: 



Let 


X be a 


non-empty set . L&1 


; P{X) denote the 


set of 


all 


subs 


ats 


of 


X. Let 


ScP(PUJ), Suppcw 


se there exists an injection 






f: 


X-*P(X> 


such that, for any 


«-« pCp(x)), <rta 


if and 


only 


if 




3x«X such that o'=f[f(i}J a 


{t(7)\ jtfW} . 










Th« 


n f is an Inacape of 3. 













This notion s rises naturally in studying the action of a 
symnlectie polarity in a projective space , Oaa of course 
wonders whether It has arisen previously in any other context. 



$t*ven H. Cullinano 

Groyp .ac.orea i Jfroblera. December 27* 1982. 

Definition: Lat G, be a finite permutation group. 

Represent <k >9 t group of permutation matrices over GF(2), 

febe tno-sl8TOBHt GflLoia field, and let V^ = ^t^ * denote 

too enveloping algebra* of £k, Suppoao there e*lst aubelgebr&a 

V 2 and V, of Y^ such that 

a] V 3 is * tranevoraal of the additive cossts of V 2 in 7j ■ and 

b) 7] li tno enveloping algebra of a subgroup G^ of G^. 
Then (& 1P V lf T 2 , 7^, G 2 J is a KroUo faQM> 

Example (Light and dark represent O's and l's): 

D0HDDDQD 

HMBBnoaca 
HacacBHcn 

EHHHMHHH 

Thla array A of matrices is a group acore In whicb 

7^ A, 

7, = the first row of A, 

7, = the first aoluawi of A, and 

Q = tha permutation matrices in the firat column, 

Hote that In the example neither 7 nor 7^ ia an Ideal of 7^ 

Problem: YEiat group score a exist? 

* 3o e Flath, AWS abstract 797-17-&6. the related 7^7-2*>-130> **li 
H. ffoyl, Tne elasalcal groups. 2nd ed. (191+6} P Princeton, p, 79 



10 



Staven H. Culllnuns . -- _- fl 

Decompositions of group enveloping algebras . Query, Hay 31, iyo.j. 

No tat 1 on i 

Lot G be an abstract group , E a subgroup of 0. Let /t:ft+M be 
a representation of as a group tf of invortible endcunorphlsme of an 
R-module V, idler* R in a conuutativo ring with unity, and let B*ye(E). 
Denota the enveloping algebra of It (i.e., the R-linear oloaur* of M) 
by S{U\, or, rare explicitly, by (E{M),+ , ■ )• For a,b In £f(U) lot 
ta,b]^ a*b - b*a, and denote the resulting Lie algebra by (E(M),+ , CJ J. 

Query i 

1, How can we relate decompositions of (E(M),+ , - ) 
to the structure of G? 

(In particular, when can wo write StM) as * direct sub 

where A is a aubalgebra of E(M)?) 

2, How can we relate decompositions of {E(X) f +,CJ ), whore T - U or N, 
to the structure of G, when M Is nonabellan? 

(In particular, ho* are tfc -e Levi direct stud decomposltlona* 

E(lE) = H(U) + L(H) and 
£(V)=a(II)+ L(N), 

where R(X) Is thB radical of {E(X],+ , £3 ) and L(X) = E(X)/fl{X) 
is a sonrialnrple Lie subalgebra of (E(X) ,+- , C3 ) f related 
to the structure of &?) 

3, How should we restrict the natures of Q, H, p , It, K, and V 
in order to answer (1) or (2) above? 



* See h, Ia Hal'cev, On semisiji^le subgroups of Lie groups (19Ui) • 
AMS Translations, series 1, volume 9 09e2) . 



11 



Steven H. CuHInaiie 

An Invariants of flyimsetry. 



3s search note. June 21, 1983. 




B 



Theorem: There exists a triply trana3.-tl.Tr* group & of 

1,2^0 ,157, k^k-i^k^ permutatiotia of the 6fy subcubea of B such that 

every G- image of B has a rigid-motion symrastry, 

(The marking on each eubeube of B is Identical; Bach la symmetric 
tinflar reflection in itfi coivfcay. ) 

Proof (sketch): ffa label the- 6k noils of B with the points of the 
afflne 6-apace A over (JF(2) In such a way that each hyperplasia of A 

is left Invariant or ±3 carried to ita complement, under a group C 
of S rigid motions generated by reflections in old-planes or B, 

We then dafina the group 0- &.Q tha group of afflne transforation a of A, 
Under Gj ae under Cj the set of bypsTplanes Bind. hyp erplane-cojiipl amenta 
is left invariant. This aynmietry of hyperplanea la then fairly easily 
shown to underlie the remarkable inTarianea of sjaaustry of B* 

(For a geometrically natural way to generate & aee £113 abstract 79T-A37.) 



12 



Stfliron H» Culllnane 

Group Identity algebran. Problem. August t|_, 1*^3. 



Definition: Let (3, »} and (S, o ) be groups with the jwu« sat S 
of element' symbol a but with different group tables. 

If there is at least one nlgabraic Identity I expressing a nontrlTial 
relationship between * and ° then fS > *, o } ia a sort of algebra, 
which for lack of any other najna we call a group Identity algebra. 

Exajmple: Let 5 z {e,a,b,o'J and let * and * be the operations 
+- and * {or Juxtaposition) In the table e below. 



+ 





a 


b 


c 


a 


e 


a 


b 


c 


a 


a 


• 


c 


b 


fa 


b 


c 


a 


a 


c 


c 


b 


a 


a 



■ 


fi 


n 


b 





o 


B 


H 


b 





B 


a 


b 


a 


a 


fa 


b 


a 


a 


a 


c 


c 


A 


Q 


b 



The following identity I holds. V ijiijirt (e>ajb»e\j 

(txy)+ (™}) + (U + y)U+ w)} - 
(Us) + (yw)) + (U + i)(7+ *)) = 
(<xw) + (yz)> + (U* w)(j+*]>. 

The dual identity I 1 obtained fay interchanging + and * 
also holds,, 

(Note that in thia case I and 1' state that certain 
algebraic forma are Invariant under the action of the 
symmetric group on their lndetermlnates. ) 

Froblea: Are there infinitely many finite group identity algebraa? 
(Mote the word "nontrlTial" in the definition. } 



13 



+ 


a 


a 


b 


c 


e 


o 


a. 


b 


e 


n 


ft 


e 


c 


b 


b 


b 


c 


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a 


= 


o 


b 


a 


o 



- l a a b c 

6 • a "b o 

a. a b c e 

b b a e a 

c c e a b 



Steven H, Cull inane 

Transformations ever a bridge . Problem, August 16, 19B3« 

Lst [G, + , * J bo the algebra (In the sense of universal algebra) 

with underlying aet G^(o,a,b,c^ and operations as follows. 

(.Re follow the not at I o rial 
conventions of writing * as 
juxtaposition and letting * 
precede + to avoid a 
proliferation of parentheses,) 

{Such an algebra (G,# j*)j where (0, *) and 1<J,^) aro groups, 
we call a bridge . } 

Problem . 

Part _1 . what is the nature of the group T generated by permutations 
of Gxfl of the form t(p,q,r,ft): lx,y) -*■ (px + qj* rx+ sy) -+- (x»y) 
nhere F,q t rjS 4 T 

Note 1.1 . It appears that T Is Isomorphic to a subgroup of 

the group of regular affine trans formations of the afflne 

|j- apace over GF(2). 

Hote I.E . we can have tCp^q^r, a) =: t(t,u,T>w) 
where fp,q#r,s)^ (tl,UjT,wJ 

since, for instance* V(x*yH 6-xG, ajc+ay = cx+cy. 
Part 2 . Is there some reasonably simple algebraic expression 
over (0*+,') for t{p,q,r,s) a t(t,u,v,ir) T 

Hflte a.l . Hot every member of T can be written in 

the form t{p,q,r, s). Example: (tfe., c*o,e) ) ♦ 



14 



Steven H. CulUQftne 

Portrait of 0. Action of tlio octaiwar&l group on a diamond. Oct. 1 'Hj, 

n^XN virfiay pxsy^ *ft^<p asv^ «>cw 

MX+D yfc^V! VS?A» ^^2VKA«i>« 

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sz;y£> a^fS^ DHti Asayt<HKiR(S'« , a 

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»«y^i?«^ft I4MQ ^2AOy]«^^ 

^V^Si ♦MXO *C^^^ >«<M *^K S2VA 

2sv^ aiHi *fi^y^ c<<ki> »<b<p;v avsjs 
*<<s?s^ M + ax &**?«!& w>r««: tf**3> 2SAV 

^^ID*M^«iy <T<>H ^K^tf VA32 

ft?W^(!? t^>^< VffSA 04MI £^ft ATVifc^ 

**d>i& >C<<H 5JVA3 ♦DIMV^««^9l 
d"&«* 02>H AS5V XM4D »a«9 ^^^^ 

%«s>«* w<m> SAV7 nrn* fX&w ^y*c« 

SVEA fc**d? HHX !^««y N04X ^»V» 

2asv *<$>*&* xww v^ifcrt: ♦run va^ 

VSA2 *<&tf* MHO ^>^V3 HMI^ ^<£AV 

>wcK75VAifi%^^ v^y^ yj>^y ♦Max 

[K>WVA2S»<?^ b?2K*(S f<?y^^ ni4H 
HXM52AV^»^ *S«aS> >^^^ M4ID 
-CT01> AVK? iP*%W ^i?^ ^iC^SC IDMI 



15 



Steven E. Cull Inane 

Study of . October 16, 1983- 

Th» 2\± two -colored 23t2i2 aiibufl "below represent the element b of the 

octahedral group 0, which la viewed as acting in the nano way on each 

of the 3 flubcubea of any given 2x£x2 cube. The arrangement of the 

2lj. colored. cube a may be of some interest for Its combinatorial 
properties. 



"\ /'■ ■■■' A / ■ ■ \ / 

/..... V... ./. .-.. . .v.. / ■■■- . ■ 

L_\' ^ ._'.V_._ ■ v 



■A." -..■ 




V 






16 



an fl. Culllrtarte 
Compound groups. Problem, November 6, l?83i 

Definition: Suppose a finite group G of order n cm b* represented as 
a group of permitatlons p., p 2 , • *■• Pn OQ m objects* where m< n. 
Suppose further that we can take th«se m objects to be distinct 
elements g,, Spi • -•* E^ *»f Q la such 4 *ray that the n products 

■ • 

(p n ts 1 J)(p a (« 2 )J ... (Pnt^n 

are all distinct^ i.e. constitute all of G. 

If s'.icji a group G, confounded In this way from a of Its olements, 

exists, we call Q a compound group. 

Problem: Which (If any) finite groups are compound? 



17 



Steven S. Cull Inane 

Group compounda* Problem. November 10 i 1983- 

Definition: Let P be s group of n permutations on a finltw 

m- el anient get X., 1st G be a oaQtlpl i cativo group , end let 

f map I to O, 

Let r~{p i l^l^ol , let X = (k.: 1£ l±n\ . 

Derino ^sF^Obj ^(p x )= t ~jT^ r ( Pi ( x }). 

Tiio etructure [?, Bj f t tp } la a group compound. 

Problem; 

(1) Let Q be given. Fop which P and f in f[?) a coaet of some 

subgroup of G7 When is ^ a durjectlon? 
f2) Let P be glYen.« For which (J and f do Inverse images under <J> 

form a coast decomposition of FT When is ^ an. injection? 



18 



Steven, E. Cull Inane 

Table group a. . Froblam. NtfTercbor 27* 19fl3» 

Ha regard the (unborderod) tables of groups of order n ta men irraya 

over the symbol* 1,? n In *hioh tb* first row (read from left to 

right ) 1b the some no the first column (read froa top to bottom ). 
(The entry at top left represents the Identity but need not bo the 

;:y:-t ol i.l Thus na regard eaeb of the arrays 

1 £ 3 U 3 1 z «, 

if l tt 3 And 1H2 ad a table or the four-group. 
3 1(12 2 £ 3 1 

UM Ul 3 

Such a table la determined (since It La a Latin square] by the 

entries lying below th* first row and to the right of the first coluim. 

Call this (n-l)x(D"l) portion of a group table an n-hox. 

(Note that the number B of n-b cocoa id In general greater than nllf, 
where N in tbe number of noniaorcirphlo group* of order n. For instance, 
forn-Uwehav* H^2, but B= Ci|I>(U> ratbar than (b,n(2).) 



For a given n M we may be able to sea something of how t.i.o Taricua 
order-n groups are interrelated by studying group actions on n-bojtes. 



Definition: Let 0(n} denote the direct product of (n-l)^ copies of Sq 
and regard tko components ot an el en ant e of 0(nj a a arranged In a □ 
(n-UxTn-l) array. Such .in element g acta COMponenUwlae la the obvious 
way on an n-boj to jleld art array that may or T»y not be an n-box. 
Suppose there exists aoiue- subgroup T of G(n) such that T la transitive 
on some sat B{T) of n-boKes that includes H n- boxes representing the N 
distinct (i.e., pairwise non Isomorphic) groups of order n« [We do not 
require that B(T) be closed under the action of T, nor even that 
each T- image of a member of B(t) be an n-box, ) We call such ft T 
a table group for a. 

Clearly for aaeb n there Is at least one table group T, namely G(n). 
That smaller T'a nay exist is shown by the following. 

Example: A table group for Jf is generated by the following 
elemsnta of ij(ij ), 



I (12) (IS) (13) (13) (13) (It) tite Ok) 

(12) (12) (13) (13) I (13) (14) tit) (Ik) 

023 (12) 112) (13) (13) (13) lm W r 



Problem; What is the order of a sn.nl 1a at table group Tor n? la there 
some way to construct such group a that does not require knowledge of HI 
(The oasa for n a prima power seems of particular interest.) 

41 
Tha example is of course not a smallest table group for U, 
but is shown for its structural in to rest. 



19 



St von H- Cullinone - - 

Linear operat or-a In geonmtrle function apace* . Probloja. Jan. 5» 1?3U- 

Let X to a 2n-4ltaatialonil llrmar apace over ft f l&ld IT. 
Lafc ft map o tnke *»oh subepaofl 5 of X to a. function f £ : X ■ »■ K 
that la noiisaro on S and zero els* whore. [3ere fj 1j t *ort of 
char act oris tic function repreaenttng tha fl^ibspnoe 3, } Denote by 
P = P<£n, K, o) the linear Space o»er K spanned by the function f^. 
1**9 call F a aBomatr^o funotlem apace. 

k 

Theorem; There la at least one P(2n, It, c} for whloh there exlata 
A iineax operator T, ncLing oil F, auth that T r taltsa the 1 -dlraan^ lon.il 
auhipacon of X (i.o., fun at ions fa representing such aubspacnaj 
to distinct r*l -dimensional gubspacoa of X, for l^r£3m-£. 



Proof; The matrix V at rdght 

r ?j :.■:- 1 ■:.■:• 3itJ SUoh alt Operator wbetl 
X la the linear 1;- apace ovor {jF{£), 
th» two-elftmeat Galois field. 



X: 



T: 



titiesta | l | l i l i l 

4 e ft 6 1 li I It 1 i " C- c £ 

#41 i a d t I m n M I t 

11 UHUDI I 4aM< 



9D 

I I 0. 

j D 

1 O 1 

I i a 

J I Q 

I e o 

D 

I 4 9 

I ]■ n 

1 a ft 

i a « 

1 a 1 

I o O 

I e o 

l e> ■ 



ceo 
etc 

c id J 

etc 

v l 

1 d ft 

8 Q O 
DIB 
I I 
□ at 
oca 



4 a 
hS tft 
ISO 

-•• a i 

Cog 
o L o 

« a i 

4 
a e e 

4 a 4 

g t a 
t a 6 

C I D 

1 o o 
ado 



boa 

e o 

1 »o 

e i a 

I 4 

DO 

1 9 O 
SO) 

a I 

til 

4 4 4 
C | 4 
04 4 
4 4 
o O I 
ft O-O 



od « a 

40 4 D 

cool 

a a a a 

I ItC 

note 

o a o a 

04SI 

00 I o 

1 a » l 
Olio 
a 44 e 
« o a a 
i o a 

1114 



ProMem: For what other apaeoa F{2n, K, c) does &uah a T oxlatT 



20 



Steven H, Cull inane 
Mamcmda and whirls . 



Expository note. Sept. 1*J, 198tj_. 



Modulo col or- interchange, and. rotations, thora are exactly 2. ways 
[aee fig. 2) to color the 6 faces of a cube ao that 

(a) each face la split diagonally into a black half and. a white 

half, and 
(d) there are exactly (j distinct iaafiea of the colored cute 

under tha group of £l|. rotational wyOTOfitriea of the cube. 

The rotational aynaastrioa of eacb anch. coloring form an ardor-S 
subgroup of leaving invariant an innoribed hexagon as In fig. 1. 

ThLn subgroup of oonaiats of the identity, potations of 120 and 2^0 
degrees about a diagonal of the cube, and lQO-degraa rotations about 
each of 3 axoa Joining midpointa of opposite adgse of the cube. 





Big.. 1 



Fig. 3 
"Diamond" and "whirl™ cubos 



Identical copies of these cubes, variously oriented, can be 

assembled into larger cubical patterns with remarkable symmetry 
properties m, 




Pig- 3 
A: Eight diamond cubes B: 




Sight whirl cubes 




Patterns A and B in fig. 3 yield a number of other symmetric 
pattern a when, their sub cubes are permuted (without rotation} aa 
follows. 

Let Sij_ act on the Ij. 1x1*2 "bricks" in each of the 3 partitions 
above; thB group A so generated can he shown to be triply transitive, 
of order l3l|ij-» and isoaiorphio to the affine group on the linear 
3-apaoo over the two -element finite field* 

THEOHiS^; Patterns A and B each have 168 Images under A„ Each of 
theae images has some nontrivial symmetry (ordinary symmetry for 
A-imagea, ordinary or color- interchange symmetry Tor B-inages) under 
at leant one of a group of ^ rigid motions of the cube, 

© i<y8j + sac 



21 



atflven H. Cullinaiuo 

Affln© groups on flnaH binary spaces, 



EipoBitoiry note, Septi 2.$, 1?9L|., 




1 



Thaorom: 

Hi* afflne group 

ASL(3j2) 

AGL££,2) 

AGL(6,2) 



or order 

322,560 

319,979,530 

1,290,157,^,^ 



is generated by 

a acting on partitions 

A 

A, B, G 

A, ^ 3 

A, B, C, 2, 3 

A, Bj Cj 1, 2-i 3> 



22 



Steven R. Culllnajne 

Visualizing fjL(2jp). Expositor? noto. Karch ?6, 19o?. 



"The typical Axiucple of a finite group la 0L(n,q}, 
tJiB general linear jp'oup of n a linens iond over fhft 
field, with 9. elenjertts, 11 -- J. L. Alpevifl. 



et> 






# a 



(ii) &) (r:j $} 



ill ^ 



: 




« 



arj es) 



(ft) 



{£) 



C*8 






**— *■ 



00 













tit) ap (n) so 




j* — *■ 



'1 



(S) UO (!?) W) { 



tf) 



(S) 



(f£) W 




(to) \i P ) 



J 



j 



W) (*) W 80 



4 »- 



CSEJ 



Q S 



tii) s) (a) Ci) (?:) (K) 



& m \m 



29 


& 


(SI) IT!) 


52 6? 


(ii) (9) 


)K '# 


(i?) tfl 


^ 


K 


{it.} \Lh) 


m 


t5d 


&.) (;:) 


£2 





Cli) (2) ea 



Tha 1^8- actions of GLi3,33 °rt a 3*3 eoordinate-nrray A ara 

1 llufltrotcd abdv*, Tho matrices shown right -multiply the slomnnts 

of A, wh«™ (1[|1 flilMl-t , 

Actiona of GL{2,p) on a pjcp co ordinal ft -array have, the sama aorts 
of symnotri'Sa, *h©ra p 1 * <iny odd prima. 



23 



Stovso H, ffulllnane 

OUgjj) aatl&na Ott a cube. Expository not*. April 5, I9QS. 

Tbo 1$ diagram* halon illuatrat* aomo 97101110 tr la ■ of ail2 t }1 action* 
on tha 8 r,anirtro vectors of tha linear 2-spoca evti tha 3-clQ»ent del 
Tba vectors ana vlawed as laballng vertlsaa of a cube (plctuj-od aer» 
with & Blight dlutortion, to avoid ovarlappJLng llnea}. 

Tha 41&grui9 may have aoma hourlHtlo Tallin for th* Stud* of groups 
ganwatod by mljilag 0L[£,3) eotlona with those of other groups. 




24 



5toV*Q B. CUlli P£ no 

Qrcmp aetlopa on partitioned Problem ana query* April 5p 196?. 



Two wayo of partitioning & 72-aet: 





B 



Definition; Let a be the ^roup of degree 72 generated by mixing 

(1) actions of tho of fine group AOL (2, 3) on the est of 
nine 2x2x2 oubea In partition A a 

(2) like actions of AGL(2, 3} on each of the eight 3*3 sections in H, 

(3) actions or AGL(3,2J on the oet of eight 3*3 sections In B, and 
{!(,) lite actions or ACL (3, 2) on each of the nine 2x2x2 cubes in A. 

Problem: What Is the order of 0? 

Query! Clearly many similar problems could be posed* 
What results or methods are known? 



(Hota: nmnj equivalent coordinate ayatems for the arfUw actions 
above ara available via natural nappinga of the respective linear 
apneas onto 313 <?r 2x2*2 arrajs.) 



25 



Steven. E. Culllnana 

Generating tho acta a generator. 



Expository note. April 28., 1905- 



<3 


1 


% 


*+l 


*£ 


x^l 


**+* 


TtVx VI 



m 



*=? 



cioin) tn>n i 



saic 



fi', 14 



BO I t 



*1 I I 



f 000 



1*1* 



Olo I ' I I oo 



Hlo 



IQOl 



IQl 1 



lltl 



LIU 



;s 



m 



§3 






A Sing Of 7-cyelQ The linear k.- space A Until" nsap S^ on. L 
3-l on GP[B) L over GF[2) ( = £ copies of 3 L ) 



S^ and Sg a-otln£ on row 1 heloif yield the Miracle Octaa Generator [3] : 

D MHBB 



a 


SBH'S 


B 


HHSH 


a 


HI3DQ 


s 


3BBB 


a 


BEEH 


B 


BBac 






£ 






D 












1 ■ 




■ ■ 








□■£■ 










I -a 



P^ 



Ha 








■° II 







Anart from it a use in studying the 759 octada of a Steiner aystem 
S{5jG,2l|_) — and henco tho Mathieu group W^ — tba Curtis NOG nicely 
llluatrates a natural correspondence C [Conwoll [2]j P> 7 1 totwaoa 

(a) tie 35 partitions of an 8-aet 8 

(bus!i aH G3TC8] above, or Convrell's 8 "heptads") Into two k.- set a, and 

(b) the 35 partitions oJT E Into. Toiir parallel ftfrltte planes. 

Two of tho H-pcrtitions have a common refinement Into 2-sete iff 

the same Is true of tho corresponding L-partitiona. (Cameron [l]j p. 6G) 

Note that is particularly natural in Ton 1, and "that partitions 2-5 
in each row have fl itni i a-p structures. 

1. Cameron, P.J., F&r&llaltgraa of Complete Deslanaj Camb. V. Pr. 197 k, 

2. Comrell, G,M. , The 3- apace PG{3j£) and Its group, 

Ann. of forth. 11 (191Q) 60-75. 

3- GurtlSj H,T., A nair tiomnlnatnrial approach to K^, 

Math. Prflc. (Jamb. Phil, 3oc. 79 [1976 J 25-l|5. 



26 



Steven H. Dull inane. 

3ymmetry invurlanee tinder %?- Expository note* Aug. 22, 19&5. 

The qulntuply transitive HatMeu group M^g might be expected to 
thoroughly Bcramble any neat pattern it acta on* However, recent 
work by R. T» Ourtia and. J. H. Conway [ll has the following 
remarkable consequence. 

Theorem! The set of 7 infinite plane patterns be loir la Invariant 
(modulo rigid mot tone of the plaUB, and col or -inter change) under 
Gurtl a- Conway BL- actions on the Lpt3 motifs shown as quad rant a. 





Mote that each pattern has nontrivial aynmiBtry, modulo color- 
interchange. (The motifs are ? of the 132 hoxads in an S(5*6#12] 
ingeniously constructed in tlJ •) 

REFERENCE 
1* Curt la j R. T.j The Steiner system 2(5*^*12), the Wathiou group 
H- 2 s^ul the "kitten, " Computational Group Theory, &d. iHchael 
D. Atkinson, Academic Preaa, 19&ht 353- 3^9 • 



27 



3. H. CUU-INAKE 



11/17/85. 



Pinltfl groups of the fiaiaa order are aoajotimoa related by 
a nontrlvial identity* 

Example: 



4 


a 


a 


b 


a 


Q 


e 


a 


b 





a 


a. 


a 


c 


b 


b 


b 


c 





a 


c 


c 


b 


a 


B 



* 9 » b 6 

a a a b c 

■!1 Fl b C O 

b b c 6 (i 

c c a a b 



I'i'o hare, V if, x, j, -a -'.[fi, ft, bj el , 
{DJ X*(y+a) — I1-7J+ (**=}+ ij and hence 

The dual identity I' obtained by Into ^changing * and * in (I) 
also holds, 

Such a atructur* •*» two group a joined together by a nontri^ial 
Identity «-- might ba called a "bridge." Are there Infinitely 
Many eorts flf bridges? r ass grateful to S. Comer for tha 
following reformulation of this rather vague question. 

Definitions : Let B ={(<?,#,- ]i ((/,*) and (0,-) are groups} . 
Por ft subrariaty V« H let A denote the set of identities holding 
in [(V) f«P all (G,*\,*}eV* Similarly, define A 4 . Par any set 
of identities A in the language for S let V(A ) denote the variety 
of all members of B that satisfy A « Call a variety V reducible. 
if V=V{i H )fl Vl& m ). 

Problem : Are there infinitely many irreducible eubvarletiea 
of B7 



28 



S. B. Efullijiano 

Dyn&mlo and algebraic ooopetlbility of groups. Deo. 11, 19fi£. 

(A) Observation — Mo&lsonorphlc order- a group a, each 

transitively '.permuting, the asuae n points, isay generate 
& group smaller than A . 

Example — The four group ana fr^ acting on the parties a 
of a square j gone rate Dk. 

(B) Observation — ; Hon Isomorphic crder-n groups ara sometimes 
related by a nontrivial Identity. 

Exaraplo ** 



i 


c- 


a 


b 





o 


e 


a 


b 


e 


ft. 


a 


e 


c 


b 


b 


b 


c 


a 


& 


c 


c 


b 


a 


o 



ii 


o 


n 


b 


c 


e 


o 


a 


b 


e 


a 


a 


b 


g 


s 


b 


b 


t 


* 


ft 


c 


a 





a 


b 



wtth x*(y+*J= <x-y)4- (x»i)+ * V i,y,a cta r a,b,c} 



probleais: 



(A) For which tn,k} are there k nonlaomorphic order-n groups Gj 
Leach nlth the sans elements and the same Identity element ) 
and regular permutation representation!! i\ such that 



Kfat^K.^.r^^j)^!^! i 



(B) 



For which tn,lc) are there k nco is amorphic order-n Rreupe G# 
(each with the same elements and the same identity element) 
all Interrelated by a nontrlviel algebraic identity? 

{") For which {n,k} are there aolutlona to both {A) and (B)? 



29 



S. H, Cullinan& 

0*On*trj of partitions II . Problems. Jaiiuary 11, 19flo, 

Pcflnlt^Qftg : 

liiTon 0* attR, and a finite (or countably infinite) aequance 
i, =(o-, a ? , »..) of poaitlTe real mtttiora such that Sa ; = t 
(op flueh that the partial 3UIsa Of K COUTSTge to a), call i o- 

partition ofa. Let L(0 b* tho following eurfaoe; 

LU ) - lUjjW^l** —* (**>*- TE.W*;Y* J. 
Thus L is i napping that lata Ua represent partitions by aurfoeea. 
(If the partial sums of t dlTarge but the o or ro spending eurf&oee 

converts, one might da fins L( E. } to be the Halt aurfaoa.) 

Theorem (Hieomaehua-Baohpt) i 

The 4nirfnco:i L((l, 2, .,., n) } all Interaeot at (1,2,3). 

Problems i 

1. Do any other "natural" Tamil lea of part it lone yield 
Intersection theorems of a noutrlviel nature? 

2. Kott do families of Infinite- aerie a partitions behave undei< J,? 
(Pop example, £ s = (1 t 2~ r ..., n ,-..}, Tor s>l.) 

3. la the generalisation of L by taking (*,y,z)« C 1 
l^.OnslMT difficult? 



30 



S. H. OullInenE 

Inacapea ill: PG(g,lj.) from F3(3,2). Expository note. Feb. l^ 1986. 

■Thie note suggests a way to visualize the finite geaisetriaa recently 
described by A. fieutelspachwr in an excellent expository article Lll • 

Noto.-fei.oil -- hexadecimal characters for the 15 points Of FG(3,2)l 

1 ~ 0OQ1 If = 0100 7 = 0111 A * 1010 8 sb 1101 

2 = UG10 5 = 0101 8 = 1000 B a 1011 E = HID 

3 = 0011 6 = 0110 9 - 1001 3 = 110O F ~ 1111 . 

Pacta about P5(3,2h the projective 3-spaee over GF(2) ; 

(A J Bsch of the 15 points the.j bs exproased a.a a sum of a unique 
pair of points from the set 3 - ■{l,2„3ti|-»6,C] . 

(B) Fifteen of tlw 35" lino a of FG(3,2) are distinguished by the fact 
that their pointc arise from partitions of s of the form 2 + 2+2- 
e.g., 3 =- il J 27Uf3,l l .1tfia,cj ylalcla the line i 3,^j7 1 = 
£1+2, 6+C, 3<-k} . (The remaining ?0 lines arise from petitions 
of S of the form. 3 + 3, "by summing pairs in the 3-aota.J 

to; Si* apreada, eaoh consisting of 5 mutually skew (i.e., disjoint) 
lines, asm be f craned from the 15 dlBtlnguiahed linos in (B). 

Zheae faote can be expressed graphically as follows. 



_l — 1 1 1 1 1 1 Lj — L 

1 % 3 



SI 



II" I z 1 



IcD 18S Iqe jSQti 



5PO 



= D E F 

w 

Points of PG-13J3J. 
(Note eymmetrlo- 
difference iwaa . J 



□Ens 

□BC3S 

iqe sac 

SftF 56S 4TF ft*A 

(B) 

15 distinguished lines. 
{Celled an "Inscape" 
bo cause of part- whole 
relationship,) 



Ears 

D 






1HT *H6 Mb H£ 



ED 

en 
annn 

The & aprestfa In (BJ . 

tlTote correspondence 
with S in {A) . } 



Beutelepacher describes a construction of KJ[2,lj.) with 

21 points _ the 6 polnta of S and the 15 distinguished lines {BJ, am! 

21 llnee — the 6 spreads (0) and the 15 point-psira (A), 

nSFmtSHCE 

1. EeutalBpacher, A., 21 - 6-15: A connection between two 

distinguished geoaetrioa, Am. Math. Monthly 93 (Jan. '36) , 29-ia. 



31 



The relativity proDlem in finite fceamatry. Steb. SO, 19S6- 

This ia the relativity problem: to fix 00 jeetiijaly 
a class or equivalent coordlnatiatttions and to 

aa certain the group of t pans format iona S modi at log 
between thera. 

— II. Weyl, The ClaHsical Groups, 
Princeton Dniv* Pr*, 191^6, p. 16 

In finite geometry "points" are often defined aa ordered 
n-tuplea of elements of a flnita {1.&,, Galois) field OF(q). 
What gaamotTla stmoturea ("frames of reference," in Weyl's tenpa) 
are ooordinatised by ouch n-tuploa? Weyl' a use of "objectively" 

seems to nioan that such structures should have oertaln objective — 
i«e,j purely geometric — properties invariant undai 1 each S, 

IMS note suggests suob a frame of reference far the affine 
Jj.-spa.oe QTor GFT2), and a ol&ea of 32£»560 equivalent 
eoordinatizatione of the frame. 

The frame : 4 )|-rl | array* 
The invariant atrnctu ra : 

Hie following set of 15 partitions of the frame into two S-seto, 

□nun 

^3 Bd Sal U 

A representative coopdinatleB.fclon : 

0000 oooi 0010 0011 

0100 0101 0110 0111 

zooo iaoi laio ion 
1100 1101 1110 1111 

TOie group : The group AGL(i]_,2) of 322, [JtO regular affine 

fcra-na fox-mat i one of the ordered J4- tuples over OP(£). 



32 



£, H. Cull Inane 

Gt-emp topologies. ProblopiB, March 31, 19o"o* 

If a group acta on a sot X f ther* is * natural olosure operation 
on subsets of X: define topological closure as cloFtura under 
G-actltsns, Then tha closed sot a (In both ana sea) ar* tha *!rpty sot, 
the G-orblfcs, and arbitrary unions of G-arMfcs, (A^ X is open iff 
A Is closed.) The result is a group topology T(O f X)» 

ttnTorfctinately, T{<J,X) la trivial if the group action, is tranaltive* 
But acta on th* power cot P(X) as well as on X, and *e nave 

X is nonasnpfcj' ** T(G,PU)> i" not trivial, and 

the --action la nontriviel e* T{G,*(X}) la. not discrete and not Tl 

{i.e., not all singletons ni* closed). 
(That a topology ia not ^ is unfortunate if the underlying net 1b 
infinite, but vary fortunate if tha underlying eat t« finite.} 

Let P =X, P n - FCF^tX)). and lot T a = T<G.r n U)) . 
Problem*: 

{1} Is thare a puraly set~ theoretic characterisation of the finite T n 
(i.e., among all oth*r topologies on P Q baHod on partitions that 
refine too cardinality partition)? 
{S} Consider tha topclogi** T Q Tor a faithful action P of G on X, 
(fi) la P alwaya determined by T , T^., ... r T n for a °me n= n{P)T 
(b) If H<G, how are thii T. for H related to tha T ± foT Gf 
it) ir x is ccuntabXy infinite, cun m regard tha minimal eloaed sots 
of f- as "natural" G-orbita on *qb» oontinuumt 



33 



S> Ht Culllwmo 

Picturing the arefllloat projective 3-flpa.c*. April 26, 198 6 > 



□mno DBoon 

BHfflB HSffflH 



BfflliB 

Tha IS points 



]HBH 



Lsfl L_J kjJ P_5 

HGHH 



QDDC 

The & spreads in A 

QCHH 
SBBB 

raaBQ 



The 25 hyperplanes 

nnnn 

DHHH 



The £ figures at left 
ahow a GjTHplactle 
polarity a; *ach point 
liefl in its oorro spend- 
ing hyperplane. The 15 
linos fixed undar * are 
shown in fig. A below. 



CDDD 
DBGH 
HBO 



□□as nnEH 



B 
The 35 lines 



moo n»nra 



HDD 







Sums of the i'-subsctu 
of A pictured in A 

anon 
DHBn 

DBBH 
□ BBS 



Sums qT the I4.- sub sot 3 
of A pictured In B or C 



CDDD 

DBBS 

Df?aa 

DDBB 



The K- T. Curtis oorre spends nee bo t ween the 35 lines and the 
2- Subsets and 3-&ybsota of a &-aat. This underlies IToj. . 



34 



A lLti«di* aasepi.** rslatad tn W-,| . Ma 7 ft, l'>"t, 

Wgur*j A.B^C BhBW tb* 3? linen or FQ(J,£); tig. A la ■ llnau* ocnple*. 



DSSH 

HEupm 
cnbn 

□HOJH 

nasa 
naaE 



nnna 

DHEH 
□□GB 
DSHD 

HHGH 
□HUB 

saee 



DDDD 
DEHQ 

DQEH 

DEQB 

EHBH 
HELBa 

HEIDI] 



B + C 



Ft put* a JL', R p ,-3' show 
and tbB 3 ^ partitions 

una 
eyas 

esse 

QISl 

ODDD 

□nnn 

onnn 
nnnn 



a. + a 

til* B.Tt. Curtis oorrenp 
of mi 3-4* t lfito t«s li 



A 4 C 



ondono* between th* 35 UnM 

*#ts a Thin undorlioa ^2h' 



B'+C 



EQDD 

QIBQ 
DB0B 

DIBH 

IBIB 
BBBB 

BBBB 

A' + B 1 



CDDD 
DBHQ 
DHSE 
DBBS 

HEED 
BEEE 

Llll 

A' + C 



35 



S. H. Cullinane 

The 2-cubseta of a 6-aat are tlie polnta of a FG(3,2). Mstj 2e, 1SB6, 

This nota Tms suggested by 
(l) A. Beutelapaoiuep'e model [13 of the 15 point a of FG(3»2) as tins 

15 partitions of a 6- set into throe Z-natu, and by 
(£) H, T» Curtis 1 a model E3J of the Conwell aorreapcKiaBnco C21 Tjatwaen 

tha 3$ linso of P5(3»a) and the 35 partitions of &n 6-aet into 

two Ti-aBta. 

If X ia a finite aet, we may regard the powor set FtX) as en 
elementary abelian 2-groTip in which addition is the sofc-thBoretic 
Bymmetrlo-dlffepwiQi* operation. Let K(X) be the subgroup 
of P(X) oonaistizi^ of and X, and let QtX3 = P(X}/k{X1. 

ffhen X ia a 6-set , the Z-aubsetfl form a subgroup A. of <i(X) T.'hosa 
noniaro dements we may take aa the pointa of a ^3(3,2), with 
00 1 linearity defined in the obTioua way. 

□SHE 

S§i§ SEEK 

A 1 

A subgroup of Q{X] Illustrating Subsets of a illustrating 

{l) the 15 2-subsot3 of a 6-set £1) the Ourtis correspondence between 

12J the 15 points of PG(3,£> A-to) and the 15 partitions of a 

6- set into three 2— Bete 
[2) a linear complex Ln FG-OpS) 

HEFEEtESCES 

1. Bflutelspaoher, A., £1 - o = l5': A connection between two 

distinguished goonetpisa, A™er. ifcth. Monthly 93 (1966) 29-l|l . 

2. Conwell, (I, M., The 3-apaco PG-U.2) and Its croup, 
Ann. of Math. 11 (1910) 60 -76 (eap. p. 72). 

3. Curtis, R. T._- A new combinatorial approach to IjUt, 
Kith. Proc. COM), Phil, Sac. 79 (1976) 25-h2. 



36 



s, r. cigiTmne 

21 projective partitions. Research nato* June 6 4 1986. 

Shown below are the Zl -point projective plane PG{£»fy) 
and Its dual. The point | (or lines) ere the 21 partitions 
of a 6-3et into disjoint seta A, B T w3iere |AJ - S or 1* 

Lines (or points) t 

QQBaHQ HHQBEQ HHESEH 5B§ 

Paints (or lines): 

caaQEis seeesq ssbhss BgB 

Foists on the ahwe Unee (or lines on the above points) : 

0300012 BHQQQQ BSQDnQ ESQ 
EBEHSia QQBBE3Q QEESQQ QBQ 

Bssasa aasses assess eas 

HSBBas gSGi@g SHSHBffl EBB 

SB@§BB BOSHES SHSBQH 00H 

The 6-aet pemmtatlon interchanging points and line a is ftom 
the lOacle Getad Generator of B. T. Curtis [1, p, 23] * 

REFERENCE 
1. Curtlfl, R. T-, A now o-ornhinstorial approach to M^, 
Hath. Frqc. Camh. Phil. 3oe. 79 (1976), 2S-UZ, 



37 



S, II. CUHinattB 

An outer automorphism of S4 related to Vgj,, June 11, 1986. 

Figure A below shoirs the 3 - subs* t 3 of a S-S&t S; figure B 
a hows tha locutions in A of tho triples of £-aubsata that 
partition £. 

nnmm DSSS 

^ e anen ^™ 

! QHBH eJCJEJU 

ddbo ssnn 

A B 

■ 
Together j A and B specify a a or re spontanea C between tha 15 

subset a and the 15 part It ion a* This correspondence 1 a : 1 m l* In 

a natural way to 

(l) n model of the proJacliTe piano FOC^A) In which. Che 21 

points {and slao the 21 lines) ire the 21 partitions of 5 

Into Hubseta a. a. -i uere 1X1 = £ or 1; 
(2} the Gonwell mapping of the 35 Uj. V W -partitions nf an 

d-aat onto the 35 lines of FO(3j3), which preserves certain 

Intersection properties; 
(3} the Rh T. Curtis "MOO" Bedel of tha Stelner ajatam fll5,{},2li.J afld 

of StaN as the model ■ s automorphism group. 

1st fia^a' exchange rows 2 and 3 In sack 3*2 picture a In A, 
una i«t c r nap a 2- subset s to C(s<)< If w& regard tha 2-a*ta 
and partition* a a transpoaitlona and products of transpositions, c 1 
JUndUcea aa outer automorphism p of S;. {In the above PG^i).), 
5g and p(s^) act in concern aa a jstroup of eolllneatlons. ) 



38 



&„ H, Gull inane 

Picturing outer sut ORorphi ana of S&. Expository note. 



Jul? 3t 1986. 



DBCB 

anats 

HQBU 

DDEQ 
□ □00 

D1QBQ 

□sqq 






□ HOD 



Iwaj l — i i/i 



541 



* H N 



1 11 * 


Ills 


III1 


ilii 


ill g 


M|]|n 



9 



^ 



Shown above are two ways to picture sow outer automDrphlHms of Sfc 
that Irnvo b^an discussed Ln the literature (6 in tl), ^ In £2)). 
In the top row, figure X shows the 1$ 2-aete in a 6-flet S, and 0,£ 
show the locations In X of triples of 2- acts that partition S. 
The second row shown the c-QTreap ending permutations. 

Each row's & ,$ contain & special 5~ m bsets : 

In the top i*ow theae f-subaeta are apreada of Unas Ln ft FG(3»«J); 
in the second row they are p&ralloliaiaa of S. Suoh S-mibaata teach of 
which can be selected in 6 waja, then nrranged in $1 ttnyo) determine 
the »J outer automorphlstaa of sj. 

[1J Conway, J* 8., Three lecture on e*captlonol groups (section 2.3K 
in Finite Simple (Jroupa, ad, Fowell and Higtcan, Academic Pr., 1971 1 

(2) J&nuaE, a., and Restrain, J., Outer automorphisMS of S&, 
toer. Math. Hcnthlj 39 (June- July 1982) , h.Q7-iao. 



39 



3. E* Culllnane 

in a cap e a IV: Inner and outer group actions. July 11, 1S86. 

Thla nota TPB.B suggeated by J. E. Conway's const ruction (1) of 
an onder*2 outer automorphism of a^. 



11 1 1 

llli 


x 


1 1 1 1 


*X 






mm dhqci 



ii 



HHEE 



Figures A and B above each show 16 pemnutationB of a 16-aet 
that generate groups G{A) and G(B), r a ape. c timely. Figure X shows 
16 subsets of a 3.6-set, The group a G-(a) and Gr(B) can act on 
figure X In two ways: by an Inner action on &ach of the 16 Ipta 
part a Indlrldttftlly, or by an outer action permuting the 16 parts. 

Theorem: Let a denote any permutation In A, and, let t denote the 
permutation in the corps spending location In B. Then the inner 
( outer) action of a on X induce b (is induced by) the outer tinner) 
action of b on X. The group G[A), antJ henoe 3(B), is isomorphic 
to Sy, and tbe map taking each a to its corresponding b extends to 
an involutiva outer automorphi sm of St. 

REFERENCE 
(1) Conway, J- II., Three lectures on exceptional groups 

(eection 2,3), in Finite Simple Group a, ed» M. B. Powell and 
G. Hlgja&n, Academic Preas, 1971, 



40